500 Ocean Currents in a Non-homogeneous Ocean 



and where C is an integration constant. The meaning of the function 0(z) is easily 

 understood, since for a purely geostrophic flow (from the first equation in XV.20 it) 

 follows 



pv = 0(z) + C. 



The constant C is the indeterminate reference velocity and the determination of C 

 can be readily seen to be equivalent to the determination of the depth of no meridional 

 motion, that is, the depth at which pv vanishes. Since for deeper layers F = 0, it 

 follows from (XV.22) 



8 



^ (PH') = 0. 



By this it is shown that the level of no meridional motion coincides with the level of 

 maximum vertical motion. Since the bottom currents are rather weak, the hypothesis 



dF 



IFz 



dF 



Tz 



allows the integration of equation (XV.22) between z and —d. Taking F{ — d) = 

 and p\v{ — d) ^ 0, the following expression for pw is obtained 



pw = J 



0(z) dz + C. (- + d) 



F(z). (XV.23) 



At the surface, r = 0, pw vanishes; the quantity F(0), according to (XIII. 27) is the net 

 convergence of the wind-driven layer and (XV.23) gives 



1 



■^ F(0) - [" jHz) dz 



The depth at which <P(z) + C vanishes, is the depth of no meridional motion. 



In physical terms the method, given in formal terms above, can be loosely described 

 in the following way. At any geographical position in the ocean the distribution of 

 the winds produces a net convergence (or divergence) of the surface waters. In the 

 steady state the only outlet (or inlet) for this water is downwards (upwards) through 

 the bottom of the frictional layer. In the deep frictionless (by hypothesis) geostrophic 

 flow, water elements will stretch (or shrink) vertically as they move towards the poles 

 (equator). The cumulative effect of this expansion or contraction added up over the 

 entire vertical column from the ocean bottom to the bottom of the frictional layer 

 leads to a vertical component of velocity which, by the conservation of mass, must be 

 equal to that induced at the bottom of the frictional layer by the winds. This balance 

 will only hold for a specific choice of the reference-level which thereby fixes this level 

 (Stommel). 



Stommel has given a numerical example for two "Atlantis" stations situated at 

 about 32° N., 50^ and 63'' W., respectively. Here the depth of no meridional motion is 

 found to be at about 1500 m; the maximum vertical velocity 24 x 10"^cmsec-\ 

 also occurs at this depth. This depth agrees well with that inferred by Defant from his 

 method. 



